Find the value of a such that the sum of the squares of the roots of the equation x2-(a-2) x-(a+1)=0 is least.

Question

Find the value of a such that the sum of the squares of the roots of the equation x2-(a-2) x-(a+1)=0 is least. (A) 4 (B) 2 (C) 1 (D) 3

Tardigrade Admin · Accepted Answer

Let α, β be the roots of the equation. ∴ α+β=a-2 and α β=-(a+1) Now, α2+β2=(α+β)2-2 α β =(a-2)2+2(a+1)=(a-1)2+5 ∴ α2+β2 will be minimum if (a-1)2=0, i.e. a=1.

Q. Find the value of $a$ such that the sum of the squares of the roots of the equation $x^{2} - (a - 2) x - (a + 1) = 0$ is least.

4

2

1

3

Let $α, β$ be the roots of the equation.
$∴ α + β = a - 2$ and $α β = - (a + 1)$
Now, $α^{2} + β^{2} = (α + β)^{2} - 2 α β$
$= (a - 2)^{2} + 2 (a + 1) = (a - 1)^{2} + 5$
$∴ α^{2} + β^{2}$ will be minimum if $(a - 1)^{2} = 0$ ,
i.e. $a = 1$ .

Q. Find the value of a such that the sum of the squares of the roots of the equation x2−(a−2)x−(a+1)=0 is least.

4

2

1

3

Let α,β be the roots of the equation. ∴α+β=a−2 and αβ=−(a+1) Now, α2+β2=(α+β)2−2αβ =(a−2)2+2(a+1)=(a−1)2+5 ∴α2+β2 will be minimum if (a−1)2=0, i.e. a=1.

Q. Find the value of $a$ such that the sum of the squares of the roots of the equation $x^{2} - (a - 2) x - (a + 1) = 0$ is least.

Let $α, β$ be the roots of the equation.
$∴ α + β = a - 2$ and $α β = - (a + 1)$
Now, $α^{2} + β^{2} = (α + β)^{2} - 2 α β$
$= (a - 2)^{2} + 2 (a + 1) = (a - 1)^{2} + 5$
$∴ α^{2} + β^{2}$ will be minimum if $(a - 1)^{2} = 0$ ,
i.e. $a = 1$ .